Quantum ergodicity on Riemannian manifolds

Quantum ergodicity" in the traditional sense deals with the question of (de)localization of eigenfunctions of the laplacian on a Riemannian manifold, in the limit of high eigenvalues. In this "semiclassical limit", it is known that the behavior of eigenfunctions bears some relation with the ergodic properties of a dynamical system called the "geodesic flow".

Quantum ergodicity on large graphs I: Regular graphs

In this talk, we consider finite regular graphs whose size grows to infinity, and discuss some delocalization results for eigenfunctions of the adjacency matrix (joint w. Le Masson). We will also discuss connections between QE on graphs and QE on manifolds, mostly through the work of Lindentrauss and collaborators on "arithmetic" quantum ergodicity.

Quantum ergodicity on graphs II : Perspectives on other models

Results on QE on discrete graphs are so far restricted to regular graphs (for which all points have the same number of neighbours). Here we will discuss possibilities of extension to other models : Anderson model on regular graphs (work in progress with Mostafa Sabri), percolation graphs on regular graphs. We will also put our results into perspective by comparing them to recent results on eigenvectors of random matrices.